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| In mathematics, a [[family of sets|family]] <math>\mathcal{F}</math> of [[Set (mathematics)|sets]] is of '''finite character''' provided it has the following properties:
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| #For each <math>A\in \mathcal{F}</math>, every [[finite set|finite]] [[subset]] of <math>A</math> belongs to <math>\mathcal{F}</math>.
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| #If every finite subset of a given set <math>A</math> belongs to <math>\mathcal{F}</math>, then <math>A</math> belongs to <math>\mathcal{F}</math>.
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| ==Properties==
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| A family <math>\mathcal{F}</math> of sets of finite character enjoys the following properties:
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| #For each <math>A\in \mathcal{F}</math>, every (finite or infinite) [[subset]] of <math>A</math> belongs to <math>\mathcal{F}</math>.
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| # [[Tukey's lemma]]: In <math>\mathcal{F}</math>, [[partial order|partially ordered]] by inclusion, the [[Union_(set_theory)|union]] of every [[Total_order#Chains|chain]] of elements of <math>\mathcal{F}</math> also belong to <math>\mathcal{F}</math>, therefore, by [[Zorn_Lemma|Zorn's lemma]], <math>\mathcal{F}</math> contains at least one maximal element.
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| ==Example==
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| Let ''V'' be a [[vector space]], and let ''F'' be the family of [[linearly independent]] subsets of ''V''. Then ''F'' is a family of finite character (because a subset ''X'' ⊆ ''V'' is linearly dependent iff ''X'' has a finite subset which is linearly dependent).
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| Therefore, in every [[vector space]], there exists a maximal family of linearly independent elements. As a maximal family is a [[vector basis]], every vector space has a (possibly infinite) vector basis.
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| {{PlanetMath attribution|id=3692|title=finite character}}
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| {{Mathlogic-stub}}
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| [[Category:Set families]]
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Revision as of 18:24, 9 February 2014
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